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b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | import OAI.NumberTheory.PiExponent.Ampleness.AmpleCurveDegreePositive
import OAI.NumberTheory.PiExponent.Cohomology.CurveCycleEuler
import OAI.NumberTheory.PiExponent.Geometry.CurveComponentsDimension
namespace OAI
namespace PiExponent.CurveCycle
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open PiExponentSeshadri.Geometry
theorem componentEulerDegree_eq_curveDegree {X : Scheme.{0}}
(p : X ⟶ Spec (CommRingCat.of ℂ)) (L : LineBundle X)
(C : irreducibleComponents X) (hd : topologicalKrullDim (reducedComponent X C) = 1) :
componentEulerDegree p L C = NumericalAmpleness.curveDegree p L (componentIntegralCurve C hd) := rfl
theorem componentEulerDegree_nonneg_of_ample {X : Scheme.{0}} [IsNoetherian X]
(p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p]
(hd : topologicalKrullDim X ≤ 1) (L : LineBundle X) (hL : L.IsAmple)
(C : irreducibleComponents X) : 0 ≤ componentEulerDegree p L C := by
rcases component_dimension_one_or_zero hd C with hC | hC
· rw [componentEulerDegree_eq_curveDegree p L C hC]
exact le_of_lt (NumericalAmpleness.curveDegree_pos_of_ample p L hL _)
· rw [componentEulerDegree_zero_of_dimension_le_zero p L C hC]
theorem cycleEulerDegree_pos_of_ample {X : Scheme.{0}} [IsNoetherian X]
(p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p]
(hd : topologicalKrullDim X = 1) (L : LineBundle X) (hL : L.IsAmple) :
0 < cycleEulerDegree p L := by
classical
let : Fintype (irreducibleComponents X) :=
NoetherianSpace.finite_irreducibleComponents.fintype
unfold cycleEulerDegree
apply Finset.sum_pos'
· intro C _
exact mul_nonneg (Int.natCast_nonneg _) (componentEulerDegree_nonneg_of_ample p hd.le L hL C)
· obtain ⟨C,hC⟩ := exists_component_dimension_one X hd
refine ⟨C,Finset.mem_univ _,mul_pos ?_ ?_⟩
· exact_mod_cast componentMultiplicity_pos C
· rw [componentEulerDegree_eq_curveDegree p L C hC]
exact NumericalAmpleness.curveDegree_pos_of_ample p L hL _
theorem cycleEulerDegree_margin {X : Scheme.{0}} [IsNoetherian X]
(p : X ⟶ Spec (CommRingCat.of ℂ)) (hd : topologicalKrullDim X ≤ 1)
(L H : LineBundle X) (ε : ℝ)
(hmargin : ∀ C : NumericalAmpleness.IntegralCurve X,
ε * (NumericalAmpleness.curveDegree p H C : ℝ) ≤
(NumericalAmpleness.curveDegree p L C : ℝ)) :
ε * (cycleEulerDegree p H : ℝ) ≤ (cycleEulerDegree p L : ℝ) := by
classical
have hc (C : irreducibleComponents X) :
ε * (componentEulerDegree p H C : ℝ) ≤ (componentEulerDegree p L C : ℝ) := by
rcases component_dimension_one_or_zero hd C with hC | hC
· exact hmargin (componentIntegralCurve C hC)
· rw [componentEulerDegree_zero_of_dimension_le_zero p H C hC,
componentEulerDegree_zero_of_dimension_le_zero p L C hC]
simp
unfold cycleEulerDegree
simp only [Int.cast_sum,Int.cast_mul,Int.cast_natCast]
rw [Finset.mul_sum]
apply Finset.sum_le_sum
intro C _
calc
ε * ((componentMultiplicity X C : ℝ) * (componentEulerDegree p H C : ℝ)) =
(componentMultiplicity X C : ℝ) * (ε * (componentEulerDegree p H C : ℝ)) := by ring
_ ≤ (componentMultiplicity X C : ℝ) * (componentEulerDegree p L C : ℝ) :=
mul_le_mul_of_nonneg_left (hc C) (Nat.cast_nonneg _)
theorem tensor_euler_margin {X : Scheme.{0}} [IsNoetherian X] [Nonempty X]
(p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p]
(hd : topologicalKrullDim X ≤ 1) (A : LineBundle X) (hA : A.IsAmple)
(L H M N : LineBundle X) (ε : ℝ)
(hmargin : ∀ C : NumericalAmpleness.IntegralCurve X,
ε * (NumericalAmpleness.curveDegree p H C : ℝ) ≤
(NumericalAmpleness.curveDegree p L C : ℝ)) :
ε * ((eulerCharacteristic p 1 (H.tensor M).sheaf - eulerCharacteristic p 1 M.sheaf : ℤ) : ℝ) ≤
((eulerCharacteristic p 1 (L.tensor N).sheaf - eulerCharacteristic p 1 N.sheaf : ℤ) : ℝ) := by
rw [tensor_euler_eq_cycle p hd A hA H M,tensor_euler_eq_cycle p hd A hA L N]
exact cycleEulerDegree_margin p hd L H ε hmargin
theorem euler_degree_pos_of_ample_curve {X : Scheme.{0}} [IsNoetherian X] [Nonempty X]
(p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p]
(hd : topologicalKrullDim X = 1) (L : LineBundle X) (hL : L.IsAmple) :
0 < eulerCharacteristic p 1 L.sheaf - eulerCharacteristic p 1 (structureSheaf X) := by
have h := tensor_euler_eq_cycle p hd.le L hL L (L.pow 0)
have he := eulerCharacteristic_iso p (moduleTensorRightUnit L.sheaf) 1
change eulerCharacteristic p 1 (L.tensor (L.pow 0)).sheaf =
eulerCharacteristic p 1 L.sheaf at he
rw [he] at h
change eulerCharacteristic p 1 L.sheaf -
eulerCharacteristic p 1 (structureSheaf X) = cycleEulerDegree p L at h
rw [h]
exact cycleEulerDegree_pos_of_ample p hd L hL
end
end PiExponent.CurveCycle
end OAI
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